In an RSA system, the public key of a given user is e = 31, n = 3599.
What is the private key of this user?

I have the answer: 3031. I am just not sure how it is calculated.

The first step is to factor 3599 into its component primes. Since
we're dealing with a small number this is not too hard (and indeed
note that 3599 = 60^2 - 1, so the factors are 60 - 1 and 60 + 1, i.e.
59 and 61). Then compute (p-1)(q-1) = 58.60 = 3480.

Now the decryption key d satisfies d.e = 1 (mod 3480), i.e. 31 d = 1
(mod 3480). Now we use Euler's algorithm to express 1 as an integer
combination of 31 and 3480:
3480 = 112(31) + 8
31 = 3(8) + 7
8 = 1(7) + 1
therefore
1 = 8 - 7
  = 8 - (31 - 3(8)) = 4(8) - 31
  = 4(3480 - 112(31)) - 31
  = 4(3480) - 449(31)

This gives us: 
31 d = 4(3480) - 449(31) (mod 3480)
     = 31 (-449)  (mod 3480)
Since 31 is prime we can divide to get
 d = -449 (mod 3480)
and adding 3480 to bring it into the canonical range gives us d = 3031.

Thank you very much manuka. Your help is greatly appreciated.